Repairing Holes Made in Older Carpet by Cats?

Our carpet is almost 20 years old, so it's not worth paying for professional repair. I'd like to replace it with tile or hardwood eventually. We've got three doorways where our kittens have clawed holes in it, when we were trying to shut them out of our bedroom, or they accidentally got shut in a closet.

I tried putting packing tape over the spots to discourage them, but they just clawed right through it. Any ideas?